We had a very similar problem with naps when my daughter was around the same age, and this is what worked for us, although I warn you, it takes a lot of consistency and effort.
BEGIN THIS ROUTINE BEFORE YOU THINK YOUR DAUGHTER IS TIRED. The main issue, I've found, is trying to get an overtired child to go to sleep. At four months, my children needed naps every two hours. I've read that this is pretty much the standard, so I'd try putting your daughter down no more than 2 hours after waking.
1. Feed your daughter and go through her normal going-to-sleep routine.
2. Put her in her crib. She's young enough that she won't be standing or rolling over, so just cover her up (or whatever you do) and leave even if she's crying.
3. After 2 minutes, or when her cries become really desperate, go back into her room. Do whatever you must to calm her down, including picking her up.
4. After she's calmed down (breathing is back to normal and no more crying) or after 5 minutes, whichever is SOONER, put her down again. If she continues crying, start again at STEP 2.
5. Repeat this process until she's asleep.
With my daughter, we'd have to go through this process for up to 2 hours, and it took about a month to really get her comfortable with going to sleep, but it worked. She goes to sleep very easily. Plus, even though this was a TON of work, it's a "graduated crying method" that worked for both of us. Just leaving a child that young to cry, I think, is unfair, unhelpful, and not nice. The method we used may take a bit longer, but it allows you and your daughter to figure each other out and to grow in a positive way together.